Algebraic Topology of Lie groups and Grassmannians with applications to Geometry

topology of oriented Grassmannian bundles related to the exceptional
Lie group G_2. Some of these results are new. One often encounters
these spaces when studying submanifolds of manifolds with calibrated
geometries. As an application, we deduce the existence of certain
special 3 and 4-dimensional submanifolds of G_2 holonomy Riemannian
manifolds with special properties. These are called Harvey-Lawson(HL)
pairs. Which appeared first in the work of Akbulut & Salur about G_2
dualities. Another application is to the free embeddings. We show that
if there is a coassociative-free embedding of a 4-manifold into the
Euclidean 7-space then the signature vanishes along with the Euler
characteristic. As a more recent application, we exhibit a family of
complex manifolds, which has a member at each odd complex dimension
and which has the same cohomology groups as the complex projective
space at that dimension, but not homotopy equivalent to it. We also
compute various cohomology rings. (Joint work with S.Akbulut et al.)